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Development of a trajectory constrained rotating arm rig for testing GNSS kinematic positioning
Accepted Manuscript Development of a Trajectory ConstrainedRotating Arm Rig for Testing GNSS Kinematic Positioning Yiming Quan, Lawrence Lau PII: DOI: Reference:
S0263-2241(19)30312-4 https://doi.org/10.1016/j.measurement.2019.04.013 MEASUR 6522
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
3 April 2018 10 March 2019 5 April 2019
Please cite this article as: Y. Quan, L. Lau, Development of a Trajectory ConstrainedRotating Arm Rig for Testing GNSS Kinematic Positioning, Measurement (2019), doi: https://doi.org/10.1016/j.measurement.2019.04.013
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Development of a Trajectory Constrained Rotating Arm Rig for Testing GNSS Kinematic Positioning Yiming Quan 1, Lawrence Lau1* 1
Artificial Intelligence and Optimisation Research Group; Department of Civil
Engineering, The University of Nottingham Ningbo China, Ningbo, China *
([email protected])
Abstract The positioning quality of static global navigation satellite system (GNSS) positioning can be assessed and monitored with a known reference point. However, quality assessment of real kinematic data is difficult because the true position of a moving antenna at a specific time is usually unknown. A rotating arm rig with one end hosting a prism or a GNSS antenna is proposed. With a prism mounted on the rig and by measuring the prism at more than three positions with a total station, the true circular trajectory can be mathematically determined and served as the reference in kinematic GNSS positioning testing. The RMS errors of estimated trajectory in Easting, Northing, and vertical components are 0.29 mm, 0.30 mm, and 0.41 mm, respectively. Six test cases are performed in different environments with the rotating arm rig placed near to a wall, near to a building and in an open clear area. The results show that in kinematic conditions GNSS double differenced carrier phase residuals are from 2.5 mm to 4 mm, and carrier phase multipath errors of relative positioning are from few millimeters up to 4 centimeters with resolved ambiguity. The test also show that the positioning quality can be improved with a higher sampling rate (10 Hz) and using full GNSS consultations, but less affected by changing speed of rig rotation. These tests demonstrate that the proposed rig can be used for validation and performance assessment in GNSS research.
Keyword GNSS validation tool • Performance assessment • High precision • Kinematic positioning • Constrained trajectory
Introduction Global Navigation Satellite System (GNSS) has been widely used in high precision positioning applications such as geodetic surveying, engineering surveying and deformation monitoring, and it has extensive kinematic applications such as positioning and navigation of manned/unmanned vehicles and orbit determination of low-earth-orbiting satellites. Accuracy assessment of static positioning can be obtained with known baselines [1,2]. Kinematic positioning performance can be tested with GNSS signal simulators, but hardware GNSS simulators are expensive. To assess the accuracy of kinematic GNSS positioning with real data, a known trajectory is needed. Many methods have been used to test kinematic GNSS positioning with some assumptions. Kechine et al. [3] use a small boat to test real-time kinematic positioning and regard the trajectory as a straight line. Aircraft with GNSS antennas can also be used to conduct kinematic tests [4,5,6]. Airborne and marine-borne platforms can provide real data for kinematic testing, but it is difficult to determine accurately the true trajectory with surveying methods, and the repeatability of testing is not as good as ground-based testbeds. Meng and Liu [7] present the usage of mini railway track for kinematic positioning testing, the precise track of railway is determined with ground-based laser scanning. The mini railway track with a pinched obround shape and a length of 120 m is on the roof of the Nottingham Geospatial Building on the Jubilee Campus of the University of Nottingham. A testbed on the Laboratoire Central des Ponts et Chaussees, named Sessyl, with a railway track is used for kinematic multipath testing in the work of Betaille et al. [8] and Lau and Cross [9]. Using road centerline as benchmark, Sun et al. [10] test multi-GNSS RTK for precise vehicle tracking in an urban area. The results show multi-GNSS RTK can achieve 3 cm to 9 cm accuracy when the satellite elevation mask is from 5° to 35°. The above research has been conducted to assess GNSS kinematic positioning with airborne, marine-borne, or ground-based mobile platforms. These track based methods have problems of accurate time tagging of observations at positions on the track, and along-track errors cannot be assessed without very accurate timing devices for the platform position with respect to the GNSS receiver observation time. We develop a low-cost and low power consumption platform with a millimeter or better level accuracy and precision. Our circular track design is less affected by the biases in straight track design because the momentum of the moving platform is not linear. A prism is installed on one end of rig arm, and at least four points on the circular track are measured by a total station, then the trajectory of rotation is determined by fitting a circle from these surveyed
points. The use of the rig assumes its base is stable during the rotation of the arm and the deflection of the end of the arm hosting a prism during survey is the same with that of hosting an antenna during test. How to assure these assumptions are not violated is explained in the next section. In engineering and industrial surveying, it is a common task to fit a circle in three dimensions. This problem is normally treated with two steps: first fit a plane to all the points, then fit a circle within the plane to all the points. Allan [11] presents a standard solution to fit a circle in three dimensions using three points whose coordinates are established from surveying techniques. Using more than three points can bring about redundancy as well as computation complexity. In two dimensions, when the number of points is more than three, least squares are usually used based on either algebraic fit or geometric fit, i.e., minimizing the sum of squares of algebraic or geometric distances from points to a circle. Geometric fit can estimate the center of the circle without essential bias, which is the leading bias term that cannot be discarded and independent of the number of points. Al-Sharadqah and Chernov [12] show the estimation of the circle radius based on geometric fit has an essential bias. Simple algebraic fit [13] is a very fast method. Geometric fit and simple algebraic fit methods are equivalent to the mixed adjustment model described by Leick [14] with condition models based on geometric and algebraic distances from points to a circle, respectively. However, the estimation of the covariance matrix of the parameters of a circle with simple algebraic fit will exceed the Rao-Cramer lower bound [15]. This disadvantage is overcome by Gradient-Weighted Algebraic Fit (GWAF) with different approximation methods proposed by Pratt [16] and Taubin [17]. Based on Pratt and Taubin’s work, Al-Sharadqah and Chernov [12] propose an algebraic fit method using a constraint matrix that has no essential bias which is ranging from 4.6% to 16.5% of total root mean square error using other algebraic fit methods in their tests. The constraint matrix is used in determining the true trajectory of rig rotation in this article. In the section below, we present the development and methodology of using a rotating arm rig for validation of GNSS data processing algorithms and error mitigation methods in kinematic conditions. Then we describe the collection of data and the use of rig for testing of six cases, followed by the testing results and detailed analyses of results. Conclusions are given in the last section.
Development and methodology in kinematic validation The developed rotating arm rig has four parts: a base, a cabinet, a rotating arm, and two platforms as shown Figure 1. The base with a motor and battery is at the bottom of the center. Above the base, there is a cabinet to place GNSS receivers. The total length of two arms is 4 meters. At the end of the rotor arm, there are two platforms. The rectangular platform is used to host an antenna or a prism, and the circular one is used to hold equivalent weight to balance the arms. A rotating rig can provide a fixed trajectory so that the ‘true’ carrier phase residuals can be calculated.
Fig. 1 An image shows the rotating arm rig for kinematic GNSS testing. The rectangular platform is designed to host an antenna or a prism, and GNSS receivers will be placed in the cabinet.
The proposed testing platform has following advantages: 1) the costs of rig manufacture and determination of precise track are relatively cheap. The cost of the rig is less than US$ 750; the precise trajectory can be surveyed by a total station and precisely modeled. The modeling method will be described in the next section. 2) The rig rotation is smooth and stable, which is ideal as a precise reference to identify positioning errors outside the track. 3) The rotating arm rig can be moved and set up in different environments, e.g., open clear environment and difficult environment. 4) When the rig is placed near to a wall, the rotation of the rig can simulate the transition between low and high multipath environments.
The following steps are taken to determine the true trajectory of rig rotation. 1. Install a prism onto the end of the rotation rig and use a total station to measure n (n>3) positions of the prism along the rotation trajectory as shown in Figure 2. The positions of the prism are measured with the traversing method [18].
Fig. 2 Determination of the rotation trajectory with a prism installed on one end of the rig.
2. Calculate the value of ( ,
, ), 1<
,
) of the measured
n points. 3. Fit a plane of circular trajectory using Principal Component Analysis (PCA) to determine three eigenvectors. The first two eigenvectors,
and
, can form the best fitting plane and
the third is a normal vector to the fitting plane [19]. The points in the 2D fitting plane is given by: =( ,
, )
(1)
=( ,
, )
(2)
4. Fit a 2D circle of the trajectory on the fitting plane by minimizing the square sum of distances from the data points to the circle:
(3)
where
and
denote the position of the circle in the 2D plane, and
circle,
,
,
,
is the radius of the
.
(4)
where
,
,
are moments, e.g.
,
An algebraic fit proposed by Al-Sharadqah and Chernov [12] can be used to fit the data points to a circle with the following constraint matrix
(5)
and introducing a Lagrange multiplier to minimize the function (6) Differentiating with respect to
gives (7)
Hence (8) Equation (8) can be solved using Newton’s method by starting the search at
, so that ,
i.e. parameter of the fitting circle in 2D space, , , and , can be determined. 5. Calculate the 2D fitting circle in 3D space with a phase angle of : (9) where ( ,
,
) is calculated in Step 2;
fitting trajectory is (
,
,
), 0<
and
are calculated in (1) and (2). The final
<2π.
As stated in the section of introduction, the main assumptions of using the rig are the stability and consistency of deflection at the end of the arm in surveying and testing. In practice,
weights can be added to the base, if a battery at the bottom is not heavy enough to assure stability or if other power source is used. If a heavy antenna, such as choke-ring antenna, is used, weights can be added to the platforms on the both ends of the arm during survey to compensate the change of deflection caused by the weight difference between antenna and prism.
Data description and testing methodology Four datasets were collected using the rotating arm rig for testing of kinematic GNSS positioning. All the kinematic datasets listed in Table 1 were collected at the University of Nottingham Ningbo China (UNNC) shown in Figure 3. The reference station of Datasets A and B was set on the North Pillar on the roof of Science and Engineering Building (SEB) shown in Figure 4(a). Datasets A, C, and D were collected on the roof of SEB. As shown in Figure 4(b), Dataset A was collected with the rotating arm rig near to a wall on the rooftop. Datasets C and D were collected on the rooftop (shown in Figure 4(c)) with a clear open environment with reference station set in the center of the rotating arm rig. Dataset B was collected on the UNNC campus near to a building shown in Figure 4(d). Javad Triumph VS and Septentrio NV receivers were used to collect Dataset A via a GEMS GS18 signal splitter. The rest of datasets were collected using Javad Triumph VS receivers.
Table 1 Details of the collection of the real kinematic datasets Datasets
Environment
Date and time of data collection (GPS time)
Sampling rate
A B C D
Near to a wall Near to a building Clear open area Clear open area
2017/3/27 08:43:02-08:44:00 2017/4/09 13:48:00-13:49:46 2015/7/28 09:50:00-10:50:00 2015/7/28 11:00:00-12:00:00
10 Hz 10 Hz 1 Hz 1 Hz
No. of observation epoch 581 1061 3600 3600
Fig. 3 A map shows the locations of the reference station of SEB North Pillar and test sites on the campus of the University of Nottingham Ningbo China.
Fig. 4 Images show the environments for (a) the reference station of SEB North Pillar, the collection of (b) Datasets A, (c) Datasets C and D, and (d) Dataset B.
The above datasets are used in six cases shown in Table 2 in order to demonstrate how the developed rotating arm rig can be used in GNSS research validation and performance assessment. Case 1, i.e. signal noise test, is designed to test kinematic signal noise by looking at the double-differenced carrier phase residuals. Case 2, i.e. stochastic model test, compares the performance of two stochastic models: simple elevation model, assuming the variance of the carrier phase error to be:
, and SIGMA-ε model [20] which is based on Carrier to
Noise Ratio (CNR). To understand the effects of the stochastic models on positioning accuracy clearly in Case 2, only L1 data is used in position estimation. Case 3, i.e. sampling rate test, investigates the effects of different sampling rates, i.e. 1 Hz and 10 Hz, on positioning quality, a liner fit position of every 10 epochs of 10 Hz data are compared with 1 Hz data. Case 4, i.e. speed of rotation test, is used to assess if the dynamics of the antenna affects positioning quality. The datasets used in Case 4, Datasets C and D, were collected with a tangential speed of rotation 0.72 m/s and 0.21 m/s, respectively. Datasets C and D were collected in an open clear area with choke-ring antennas to mitigate multipath effects. Case 5, i.e. PPP and relative positioning test, compares the positioning solution using Precise Point Positioning (PPP) and relative positioning.
Case 6, i.e. single and multi-GNSS test, compares the positioning quality using GPS alone and using full GNSS consultations (GPS+GLONASS+Galileo+BDS+QZSS). An elevation mask of 30° is used for data processing in Case 1 to mitigate the multipath effects and an elevation mask of 15° is used for data processing of kinematic positioning in the rest of the cases. The ambiguity is resolved by Least Squares Ambiguity Decorrelation Adjustment (LAMBDA) method [21] in the data processing.
Table 2 List of testing cases and testing datasets
Case 1 2 3 4 5 6
Description Testing kinematic signal noise by looking at the double-differenced carrier phase residuals Testing two stochastic models: simple elevation Stochastic model test model and SIGMA-ε model Testing the effects of different sampling rates on Sampling rate test positioning quality Testing the effects of speed of rover antenna on Speed of rotation test positioning quality PPP and relative Comparison of PPP and relative positioning quality positioning test Single and multiComparison of the positioning quality using GPS GNSS test alone and using full GNSS consultations Signal noise test
Dataset C&D A A&B C&D A C
Test Results The surveyed points and the corresponding fitting rotation trajectory of Dataset A are shown in Figure 5. The trajectory is calculated with (1)-(9). Corrections have been applied for the offsets between prism and GNSS antenna phase center when they are installed on the rig. It can be seen from the figure that the estimated trajectory fits well with the surveyed points. The RMS errors of fitting in Easting, Northing, and vertical components are 0.29 mm, 0.30 mm, and 0.41 mm, respectively.
Fig. 5 Total station surveyed points (with red crosses) and fitting trajectory (with a blue line).
Case 1 test results The standard deviations (S.D.) of double differenced (DD) carrier phase residuals and the number of available satellites in Case 1 are summarized in Table 3. The standard deviations of the residual slightly vary with signal types, and most of the residuals are from 2.5 mm to 4 mm. Comparing Datasets C with D, in Table 3, the results show that the dynamics of the antenna has no significant effect on signal quality. Although the speed of rig rotation in Dataset C is more than three times faster than that in Dataset D, the residuals of each signal are similar between Datasets C and D. There is an exception that the residual of GPS L5 in Dataset D is significantly larger than that in Dataset C. This is because there are only two GPS satellites, PRN6 and PRN9, with L5 signal and the elevation angles of the two satellites in Dataset D are lower than those in Dataset C. Galileo and QZSS data are not presented here because their number of available satellites is less than four for fixed ambiguity solutions.
Table 3 Standard deviations of DD carrier phase residuals using GPS, GLONASS, and BDS signals in Case 1 Dataset
C
GNSS
Freq. Band
Signal
No. of SV
GPS
L1
C/A
8
D S.D. of DD carrier phase residuals (mm) 2.09
No. of SV
9
S.D. of DD carrier phase residuals (mm) 1.96
L2 L5 L1 GLONASS L2 BDS
B1 B2
L2C(M+L) L2P(Y) I+Q L1 OF L1 SF L2 OF L2 SF I I
6 8 2 7 7 7 7 6 6
2.76 2.44 2.37 2.16 1.80 2.69 2.12 3.58 2.70
6 9 2 7 7 7 7 6 6
2.83 2.39 2.61 2.35 1.83 2.95 2.20 3.40 2.59
Case 2 test results A summary of the positioning test results of Cases 2 to 6 is shown in Table 4. The test results of Case 2 show that using SIGMA-ε model can improve the positioning accuracy by 5% to 10%, but the positioning results are receiver dependent when using the SIMGA-ε. This is because the weight assignment of SIGMA-ε model is based on CNR, and the CNRs of Javad datasets are higher than those of Septentrio datasets. Though two receivers are connected to the same antenna via a signal splitter, their CNRs are not the same because of differences in band limiting and quantization schemes within the receivers [22]. Figure 6 shows CNRS of two selected satellites, PRN11 with an elevation of about 26 degrees and PRN27 with an elevation of 67 degrees, from Javad and Septentrio receivers for Dataset A when the same antenna is used via a splitter. As a result, SIGMA-ε model is more receiver dependent compared with simple elevation model in the datasets.
Table 4 Statistical results of RMS error in Case 2
Case
Configuration
Dataset
Receiver model
2 2 2 2
Elevation model Elevation model SIGMA-ε model SIGMA-ε model
A A A A
Javad Septentrio Javad Septentrio
RMS Error (mm) Horizontal 5.0 5.0 4.7 4.6
Vertical 6.3 6.3 6.5 5.8
3D 8.0 8.0 8.0 7.4
Fig. 6 CNRs of GPS Satellites PRN11 and PRN27 with Javad and Septentrio receivers in Dataset A. CNRs of Javad dataset are higher than Septentrio dataset collected with the same antenna.
Case 3 test results In Table 5, the test results of Dataset A in Case 3 show that 10 Hz sampling rate can improve positioning accuracy by about 16% because more samples are used for averaging in higher sampling rate. The positioning quality of Dataset B is significantly degraded compared with Dataset A; it is due to severe multipath effects and poor satellite geometry because several satellites were blocked by the building as shown Figure 4(d). With the determined trajectory of rotation, the ‘true’ carrier phase measurement errors of some selected satellites in Datasets A and B are estimated and plotted in Figure 7. Figure 7 shows that Dataset A has large multipath errors around Epoch 500 when the antenna moves close to the wall, and Dataset B has larger multipath errors (up to 4 cm) than Dataset A. Though long tests have been carried out, we choose one complete cycle of rig rotation because the multipath errors in next cycle are very close to the previous cycle.
Table 5 Statistical results of RMS error in Case 3
Case 3 3 3 3
Configuration 1 Hz sampling rate 10 Hz sampling rate 1 Hz sampling rate 10 Hz sampling rate
Dataset A A B B
Receiver model Javad Javad Javad Javad
RMS Error (mm) Horizontal 4.5 3.7 11.9 9.2
Vertical 7.2 6.1 34.0 17.2
3D 8.5 7.1 36.0 19.5
Fig. 7 ‘True’ GPS L1 double-differenced carrier phase measurement errors in Datasets A (left) and B (right).
Case 4-6 test results In Table 6, the results of Case 4 show that the dynamics of the antenna has no significant effect on positioning quality. The results of Case 5 in Table 6 show that the kinematic positioning with millimeter-level accuracy significantly outperforms decimeter-accurate PPP in the datasets. This can be seen from Figure 8, where most of the PPP solutions fall outside the true trajectory (red line) whilst kinematic positioning solutions are close to the true trajectory. The main sources of decimeter-level error of PPP in Figure 8 are due to the tropospheric delay, satellite orbit and
clock errors. Unlike PPP, these errors are cancelled out in relative positioning with double differenced measurements, hence the positioning accuracy of relative positioning is better than that of PPP. The results of Case 6 show that using multi-GNSS (GPS+GLOANSS+Galileo+ BDS+QZSS) can significantly improve the kinematic positioning accuracy by about 47% compared with using GPS only in a clear open area.
Table 6 Statistical results of RMS error in Cases 4-6
Case 4 4 5 5 6 6
Configuration Speed of rotation 0.72 m/s Speed of rotation 0.21 m/s Precise Point Positioning Kinematic mode GPS only Full constellations
Receiver model
RMS Error (mm) Horizontal
Vertical
3D
C
Javad
2.1
6.4
6.7
D
Javad
2.2
6.8
7.2
A A C C
Septentrio Septentrio Javad Javad
107.8 4.6 2.2 1.5
60.6 5.4 6.8 3.5
123.3 7.0 7.2 3.8
Dataset
Fig. 8 Positioning results (blue) and true trajectory (red) using PPP (top) and kinematic positioning (bottom)
Conclusions This work has developed a rotating arm rig for kinematic testing of GNSS carrier-phase based high precision positioning. The method for determination of the trajectory of rotation with a submillimeter-level accuracy has been presented. Four datasets were collected using the rig for testing in six cases. The double differenced carrier phase residuals of GPS, GLONASS, and BDS signals in kinematic testing have been assessed in Case 1. Statistical results show the residuals slightly vary with signal types, and most of the residuals are from 2.5 mm to 4 mm in kinematic conditions. Also, the results show that the dynamics of the antenna, i.e. moving speed of antenna, has no significant effect on signal quality. The test on stochastic models in Case 2 shows that SIGMA-ε model can improve the positioning accuracy by 5% to 10% compared with simple elevation model, but SIGMA-ε model is receiver-dependent since it assigns measurement weights based on CNR. The test results of Case 3 show that a higher sampling rate (10 Hz) can improve the positioning quality comparing with 1 Hz sampling rate, especially in a difficult environment. The results also show that with resolved ambiguity, carrier phase multipath errors of relative positioning are from few millimeters up to 4 centimeters. The test results of Case 4 show that the dynamics of the antenna has no significant effect on positioning quality. The results of Case 5 show that the kinematic positioning with millimeter-level accuracy significantly outperforms decimeter-accurate Precise Point Positioning in the dataset because some sources of
errors such as tropospheric delay are cancelled out in double differencing of relative positioning. The results of Case 6 show that using multi-GNSS (GPS+GLOANSS+Galileo+BDS+QZSS) can significantly improve the kinematic positioning accuracy by about 47% in a clear open area. This article has demonstrated the use of the proposed rotating arm rig for validation of GNSS data processing algorithms and error mitigation methods in kinematic conditions. Therefore, the main contribution of this research is the design of a low-cost, low power consumption, high precision solution for rigorous kinematic GNSS research validation and performance assessments. Based on the designed rotating arm rig, in the future it is possible to use the rig for the testing and validation of other technologies and algorithms. For example, replacing the prism and antenna used in this study with photogrammetric targets or Ultra-Wide Band signal transmitter, or study relevant machine learning and filtering methods [23,24,25] for multi-sensor data fusion.
Acknowledgements The cost of building the rotating arm rig is funded by Lawrence Lau’s internal research grant (01.03.02.01.2033).
References [1] Lau, L., Tateshita, H., Sato, K., 2015, Impact of Multi-GNSS(GPS, GLONASS, and QZSS) on positioning accuracy and multipath errors in high-precision single-epoch solutions - A case study in Ningbo China. J. Navig. 68 (5) 999-1017. https://doi.org/10.1017/S0373463315000168. [2] Quan, Y., Lau, L., Roberts, G.W., Meng, X., 2015, Measurement signal quality assessment on all available and new signals of multi-GNSS (GPS, GLONASS, Galileo, BDS, and QZSS) with real data. J. Navig. 69 (2) 313-334. https://doi.org/10.1017/S0373463315000624. [3] Kechine, M.O., Tiberius, C.C.J.M., Van der Marel, H., 2004, An experimental performance analysis of real-time kinematic positioning with NASA’s Internet-based global differential GPS. GPS Solut. 8 (1) 9-22. https://doi.org/10.1007/s10291-003-0081-3.
[4] Sohn, S., Lee, B., Kim, J., Kee, C., 2008, Vision-based real-time target localization for single-antenna GPS-guided UAV. IEEE Trans. Aerosp. Electron. Syst. 44 (4) 1391-1401. https://doi.org/10.1109/TAES.2008.4667717. [5] Lau, L., Cross, P., Steen, M., 2012, Flight tests of error-bounded heading and pitch determination with two GPS receivers. IEEE Trans. Aerosp. Electron. Syst. 48 (1) 388-404. https://doi.org/10.1109/TAES.2012.6129643. [6] Wang, D., Lv, H., Wu, J., 2017, Augmented Cubature Kalman filter for nonlinear RTK/MIMU integrated navigation with non-additive noise. Measurements 97 111-125. https://doi.org/10.1016/j.measurement.2016.10.056. [7] Meng, X., Liu, C., 2013, Precise determination of mini railway track with ground-based laser scanning. Surv. Rev. 46 (336) 213-218. https://doi.org/10.1179/1752270613Y.0000000072. [8] Betaille, D., Cross, P., Euler, H.J., 2006, Assessment and improvement of the capabilities of a window correlator to model GPS multipath phase errors. IEEE Trans. Aerosp. Electron. Syst. 42(2) 707-718. https://doi.org/10.1109/TAES.2006.1642583. [9] Lau, L., Cross, P., 2007, Investigations into phase multipath techniques for high precision positioning in difficult environments. J. Navig. 60 (3) 457-482. https://doi.org/10.1017/S0373463307004341. [10] Sun, Q., Xia, J., Foster, J., Falkmer, T., 2017, Pursuing precise vehicle movement trajectory in urban residential area using Multi-GNSS RTK tracking. Transp. Res. Procedia. 25:23612376. https://doi.org/10.1016/j.trpro.2017.05.255. [11] Allan, A.L., 1996, Circle fitting in three dimensions. Surv. Rev. 33 (260) 416-422. https://doi.org/10.1179/sre.1996.33.260.416. [12] Al-Sharadqah, A., Chernov, N., 2009, Error analysis for circle fitting algorithms. Electron. J. Stat. 3(2009) 886-911. https://doi.org/10.1214/09-EJS419. [13] Kasa, I., 1976, A curve fitting procedure and its error analysis. IEEE Trans. Inst. Meas. 25(1) 8-14. https://doi.org/10.1109/TIM.1976.6312298. [14] Leick, A., 2004, GPS Satellite Surveying. 3rd Edition, John Wiley and Sons, Hoboken. ISBN: 978-0-471-05930-1
[15] Chernov, N., Lesort, C., 2005, Least squares fitting of circles. J. Math. Imag. Vision 23 239–251. https://doi.org/10.1007/s10851-005-0482-8. [16] Pratt, V., 1987, Direct least-squares fitting of algebraic surfaces. Comput. Graphics 21(4) 145-152. https://doi.org/10.1145/37402.37420. [17] Taubin, G., 1991, Estimation of planar curves, surfaces and nonplanar space curves defined by implicit equations, with applications to edge and range image segmentation, IEEE Trans. Pattern Anal. Mach. Intell. 13 (11) 1115–1138. https://doi.org/10.1109/34.103273. [18] Uren, J., Price, B., Surveying for Engineers, fifth ed., Red Globe Press, London, 2010. [19] Jolliffe, I., 2002, Principal component analysis, 2nd ed. Springer. ISBN 978-0-387-22440-4 [20] Hartinger, H., Brunner, F.K., 1999, Variances of GPS phase observations: the SIGMA-ε model. GPS Solut. 2(4) 35-43. https://doi.org/10.1007/PL00012765. [21] Teunissen, P.J.G., 1995, The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J. Geod. 70(1-2) 65-82. https://doi.org/10.1007/s001900050127. [22] Joseph, A., 2010, Measuring GNSS signal strength. Inside GNSS 5(8):22-25. [23] Akbarizadeh, G., 2012, A new statistical-based kurtosis wavelet energy feature for texture recognition of SAR images. IEEE Trans. Geosc. Remote Sens. 11(50) 4358-4368. https://doi.org/10.1109/TGRS.2012.2194787. [24] Karimi, D., Akbarizadeh, G., Rangzan, K., Kabolizadeh, M., 2017, Effective supervised multiple-feature learning for fused radar and optical data classification. IET Radar Sonar Navig. 11(5) 768-777. https://doi.org/10.1049/iet-rsn.2016.0346. [25] Norouzi, M., Akbarizadeh, G., Eftekhar, F., 2018, A hybrid feature extraction method for SAR image registration. Signal Image Video Process. 12: 1559. https://doi.org/10.1007/s11760-018-1312-y.
Highlights
A rotating arm rig with one end hosting a prism or a GNSS antenna is proposed True circular trajectory of arm rotation is mathematically determined by total station measurements Determined trajectory served as a reference in kinematic GNSS positioning testing Assessing signal quality and positioning quality in kinematic condition Study effects of stochastic model, sampling rate, antenna moving speed, and GNSS constellation
S0263-2241(19)30312-4 https://doi.org/10.1016/j.measurement.2019.04.013 MEASUR 6522
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
3 April 2018 10 March 2019 5 April 2019
Please cite this article as: Y. Quan, L. Lau, Development of a Trajectory ConstrainedRotating Arm Rig for Testing GNSS Kinematic Positioning, Measurement (2019), doi: https://doi.org/10.1016/j.measurement.2019.04.013
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Development of a Trajectory Constrained Rotating Arm Rig for Testing GNSS Kinematic Positioning Yiming Quan 1, Lawrence Lau1* 1
Artificial Intelligence and Optimisation Research Group; Department of Civil
Engineering, The University of Nottingham Ningbo China, Ningbo, China *
([email protected])
Abstract The positioning quality of static global navigation satellite system (GNSS) positioning can be assessed and monitored with a known reference point. However, quality assessment of real kinematic data is difficult because the true position of a moving antenna at a specific time is usually unknown. A rotating arm rig with one end hosting a prism or a GNSS antenna is proposed. With a prism mounted on the rig and by measuring the prism at more than three positions with a total station, the true circular trajectory can be mathematically determined and served as the reference in kinematic GNSS positioning testing. The RMS errors of estimated trajectory in Easting, Northing, and vertical components are 0.29 mm, 0.30 mm, and 0.41 mm, respectively. Six test cases are performed in different environments with the rotating arm rig placed near to a wall, near to a building and in an open clear area. The results show that in kinematic conditions GNSS double differenced carrier phase residuals are from 2.5 mm to 4 mm, and carrier phase multipath errors of relative positioning are from few millimeters up to 4 centimeters with resolved ambiguity. The test also show that the positioning quality can be improved with a higher sampling rate (10 Hz) and using full GNSS consultations, but less affected by changing speed of rig rotation. These tests demonstrate that the proposed rig can be used for validation and performance assessment in GNSS research.
Keyword GNSS validation tool • Performance assessment • High precision • Kinematic positioning • Constrained trajectory
Introduction Global Navigation Satellite System (GNSS) has been widely used in high precision positioning applications such as geodetic surveying, engineering surveying and deformation monitoring, and it has extensive kinematic applications such as positioning and navigation of manned/unmanned vehicles and orbit determination of low-earth-orbiting satellites. Accuracy assessment of static positioning can be obtained with known baselines [1,2]. Kinematic positioning performance can be tested with GNSS signal simulators, but hardware GNSS simulators are expensive. To assess the accuracy of kinematic GNSS positioning with real data, a known trajectory is needed. Many methods have been used to test kinematic GNSS positioning with some assumptions. Kechine et al. [3] use a small boat to test real-time kinematic positioning and regard the trajectory as a straight line. Aircraft with GNSS antennas can also be used to conduct kinematic tests [4,5,6]. Airborne and marine-borne platforms can provide real data for kinematic testing, but it is difficult to determine accurately the true trajectory with surveying methods, and the repeatability of testing is not as good as ground-based testbeds. Meng and Liu [7] present the usage of mini railway track for kinematic positioning testing, the precise track of railway is determined with ground-based laser scanning. The mini railway track with a pinched obround shape and a length of 120 m is on the roof of the Nottingham Geospatial Building on the Jubilee Campus of the University of Nottingham. A testbed on the Laboratoire Central des Ponts et Chaussees, named Sessyl, with a railway track is used for kinematic multipath testing in the work of Betaille et al. [8] and Lau and Cross [9]. Using road centerline as benchmark, Sun et al. [10] test multi-GNSS RTK for precise vehicle tracking in an urban area. The results show multi-GNSS RTK can achieve 3 cm to 9 cm accuracy when the satellite elevation mask is from 5° to 35°. The above research has been conducted to assess GNSS kinematic positioning with airborne, marine-borne, or ground-based mobile platforms. These track based methods have problems of accurate time tagging of observations at positions on the track, and along-track errors cannot be assessed without very accurate timing devices for the platform position with respect to the GNSS receiver observation time. We develop a low-cost and low power consumption platform with a millimeter or better level accuracy and precision. Our circular track design is less affected by the biases in straight track design because the momentum of the moving platform is not linear. A prism is installed on one end of rig arm, and at least four points on the circular track are measured by a total station, then the trajectory of rotation is determined by fitting a circle from these surveyed
points. The use of the rig assumes its base is stable during the rotation of the arm and the deflection of the end of the arm hosting a prism during survey is the same with that of hosting an antenna during test. How to assure these assumptions are not violated is explained in the next section. In engineering and industrial surveying, it is a common task to fit a circle in three dimensions. This problem is normally treated with two steps: first fit a plane to all the points, then fit a circle within the plane to all the points. Allan [11] presents a standard solution to fit a circle in three dimensions using three points whose coordinates are established from surveying techniques. Using more than three points can bring about redundancy as well as computation complexity. In two dimensions, when the number of points is more than three, least squares are usually used based on either algebraic fit or geometric fit, i.e., minimizing the sum of squares of algebraic or geometric distances from points to a circle. Geometric fit can estimate the center of the circle without essential bias, which is the leading bias term that cannot be discarded and independent of the number of points. Al-Sharadqah and Chernov [12] show the estimation of the circle radius based on geometric fit has an essential bias. Simple algebraic fit [13] is a very fast method. Geometric fit and simple algebraic fit methods are equivalent to the mixed adjustment model described by Leick [14] with condition models based on geometric and algebraic distances from points to a circle, respectively. However, the estimation of the covariance matrix of the parameters of a circle with simple algebraic fit will exceed the Rao-Cramer lower bound [15]. This disadvantage is overcome by Gradient-Weighted Algebraic Fit (GWAF) with different approximation methods proposed by Pratt [16] and Taubin [17]. Based on Pratt and Taubin’s work, Al-Sharadqah and Chernov [12] propose an algebraic fit method using a constraint matrix that has no essential bias which is ranging from 4.6% to 16.5% of total root mean square error using other algebraic fit methods in their tests. The constraint matrix is used in determining the true trajectory of rig rotation in this article. In the section below, we present the development and methodology of using a rotating arm rig for validation of GNSS data processing algorithms and error mitigation methods in kinematic conditions. Then we describe the collection of data and the use of rig for testing of six cases, followed by the testing results and detailed analyses of results. Conclusions are given in the last section.
Development and methodology in kinematic validation The developed rotating arm rig has four parts: a base, a cabinet, a rotating arm, and two platforms as shown Figure 1. The base with a motor and battery is at the bottom of the center. Above the base, there is a cabinet to place GNSS receivers. The total length of two arms is 4 meters. At the end of the rotor arm, there are two platforms. The rectangular platform is used to host an antenna or a prism, and the circular one is used to hold equivalent weight to balance the arms. A rotating rig can provide a fixed trajectory so that the ‘true’ carrier phase residuals can be calculated.
Fig. 1 An image shows the rotating arm rig for kinematic GNSS testing. The rectangular platform is designed to host an antenna or a prism, and GNSS receivers will be placed in the cabinet.
The proposed testing platform has following advantages: 1) the costs of rig manufacture and determination of precise track are relatively cheap. The cost of the rig is less than US$ 750; the precise trajectory can be surveyed by a total station and precisely modeled. The modeling method will be described in the next section. 2) The rig rotation is smooth and stable, which is ideal as a precise reference to identify positioning errors outside the track. 3) The rotating arm rig can be moved and set up in different environments, e.g., open clear environment and difficult environment. 4) When the rig is placed near to a wall, the rotation of the rig can simulate the transition between low and high multipath environments.
The following steps are taken to determine the true trajectory of rig rotation. 1. Install a prism onto the end of the rotation rig and use a total station to measure n (n>3) positions of the prism along the rotation trajectory as shown in Figure 2. The positions of the prism are measured with the traversing method [18].
Fig. 2 Determination of the rotation trajectory with a prism installed on one end of the rig.
2. Calculate the value of ( ,
, ), 1<
,
) of the measured
n points. 3. Fit a plane of circular trajectory using Principal Component Analysis (PCA) to determine three eigenvectors. The first two eigenvectors,
and
, can form the best fitting plane and
the third is a normal vector to the fitting plane [19]. The points in the 2D fitting plane is given by: =( ,
, )
(1)
=( ,
, )
(2)
4. Fit a 2D circle of the trajectory on the fitting plane by minimizing the square sum of distances from the data points to the circle:
(3)
where
and
denote the position of the circle in the 2D plane, and
circle,
,
,
,
is the radius of the
.
(4)
where
,
,
are moments, e.g.
,
An algebraic fit proposed by Al-Sharadqah and Chernov [12] can be used to fit the data points to a circle with the following constraint matrix
(5)
and introducing a Lagrange multiplier to minimize the function (6) Differentiating with respect to
gives (7)
Hence (8) Equation (8) can be solved using Newton’s method by starting the search at
, so that ,
i.e. parameter of the fitting circle in 2D space, , , and , can be determined. 5. Calculate the 2D fitting circle in 3D space with a phase angle of : (9) where ( ,
,
) is calculated in Step 2;
fitting trajectory is (
,
,
), 0<
and
are calculated in (1) and (2). The final
<2π.
As stated in the section of introduction, the main assumptions of using the rig are the stability and consistency of deflection at the end of the arm in surveying and testing. In practice,
weights can be added to the base, if a battery at the bottom is not heavy enough to assure stability or if other power source is used. If a heavy antenna, such as choke-ring antenna, is used, weights can be added to the platforms on the both ends of the arm during survey to compensate the change of deflection caused by the weight difference between antenna and prism.
Data description and testing methodology Four datasets were collected using the rotating arm rig for testing of kinematic GNSS positioning. All the kinematic datasets listed in Table 1 were collected at the University of Nottingham Ningbo China (UNNC) shown in Figure 3. The reference station of Datasets A and B was set on the North Pillar on the roof of Science and Engineering Building (SEB) shown in Figure 4(a). Datasets A, C, and D were collected on the roof of SEB. As shown in Figure 4(b), Dataset A was collected with the rotating arm rig near to a wall on the rooftop. Datasets C and D were collected on the rooftop (shown in Figure 4(c)) with a clear open environment with reference station set in the center of the rotating arm rig. Dataset B was collected on the UNNC campus near to a building shown in Figure 4(d). Javad Triumph VS and Septentrio NV receivers were used to collect Dataset A via a GEMS GS18 signal splitter. The rest of datasets were collected using Javad Triumph VS receivers.
Table 1 Details of the collection of the real kinematic datasets Datasets
Environment
Date and time of data collection (GPS time)
Sampling rate
A B C D
Near to a wall Near to a building Clear open area Clear open area
2017/3/27 08:43:02-08:44:00 2017/4/09 13:48:00-13:49:46 2015/7/28 09:50:00-10:50:00 2015/7/28 11:00:00-12:00:00
10 Hz 10 Hz 1 Hz 1 Hz
No. of observation epoch 581 1061 3600 3600
Fig. 3 A map shows the locations of the reference station of SEB North Pillar and test sites on the campus of the University of Nottingham Ningbo China.
Fig. 4 Images show the environments for (a) the reference station of SEB North Pillar, the collection of (b) Datasets A, (c) Datasets C and D, and (d) Dataset B.
The above datasets are used in six cases shown in Table 2 in order to demonstrate how the developed rotating arm rig can be used in GNSS research validation and performance assessment. Case 1, i.e. signal noise test, is designed to test kinematic signal noise by looking at the double-differenced carrier phase residuals. Case 2, i.e. stochastic model test, compares the performance of two stochastic models: simple elevation model, assuming the variance of the carrier phase error to be:
, and SIGMA-ε model [20] which is based on Carrier to
Noise Ratio (CNR). To understand the effects of the stochastic models on positioning accuracy clearly in Case 2, only L1 data is used in position estimation. Case 3, i.e. sampling rate test, investigates the effects of different sampling rates, i.e. 1 Hz and 10 Hz, on positioning quality, a liner fit position of every 10 epochs of 10 Hz data are compared with 1 Hz data. Case 4, i.e. speed of rotation test, is used to assess if the dynamics of the antenna affects positioning quality. The datasets used in Case 4, Datasets C and D, were collected with a tangential speed of rotation 0.72 m/s and 0.21 m/s, respectively. Datasets C and D were collected in an open clear area with choke-ring antennas to mitigate multipath effects. Case 5, i.e. PPP and relative positioning test, compares the positioning solution using Precise Point Positioning (PPP) and relative positioning.
Case 6, i.e. single and multi-GNSS test, compares the positioning quality using GPS alone and using full GNSS consultations (GPS+GLONASS+Galileo+BDS+QZSS). An elevation mask of 30° is used for data processing in Case 1 to mitigate the multipath effects and an elevation mask of 15° is used for data processing of kinematic positioning in the rest of the cases. The ambiguity is resolved by Least Squares Ambiguity Decorrelation Adjustment (LAMBDA) method [21] in the data processing.
Table 2 List of testing cases and testing datasets
Case 1 2 3 4 5 6
Description Testing kinematic signal noise by looking at the double-differenced carrier phase residuals Testing two stochastic models: simple elevation Stochastic model test model and SIGMA-ε model Testing the effects of different sampling rates on Sampling rate test positioning quality Testing the effects of speed of rover antenna on Speed of rotation test positioning quality PPP and relative Comparison of PPP and relative positioning quality positioning test Single and multiComparison of the positioning quality using GPS GNSS test alone and using full GNSS consultations Signal noise test
Dataset C&D A A&B C&D A C
Test Results The surveyed points and the corresponding fitting rotation trajectory of Dataset A are shown in Figure 5. The trajectory is calculated with (1)-(9). Corrections have been applied for the offsets between prism and GNSS antenna phase center when they are installed on the rig. It can be seen from the figure that the estimated trajectory fits well with the surveyed points. The RMS errors of fitting in Easting, Northing, and vertical components are 0.29 mm, 0.30 mm, and 0.41 mm, respectively.
Fig. 5 Total station surveyed points (with red crosses) and fitting trajectory (with a blue line).
Case 1 test results The standard deviations (S.D.) of double differenced (DD) carrier phase residuals and the number of available satellites in Case 1 are summarized in Table 3. The standard deviations of the residual slightly vary with signal types, and most of the residuals are from 2.5 mm to 4 mm. Comparing Datasets C with D, in Table 3, the results show that the dynamics of the antenna has no significant effect on signal quality. Although the speed of rig rotation in Dataset C is more than three times faster than that in Dataset D, the residuals of each signal are similar between Datasets C and D. There is an exception that the residual of GPS L5 in Dataset D is significantly larger than that in Dataset C. This is because there are only two GPS satellites, PRN6 and PRN9, with L5 signal and the elevation angles of the two satellites in Dataset D are lower than those in Dataset C. Galileo and QZSS data are not presented here because their number of available satellites is less than four for fixed ambiguity solutions.
Table 3 Standard deviations of DD carrier phase residuals using GPS, GLONASS, and BDS signals in Case 1 Dataset
C
GNSS
Freq. Band
Signal
No. of SV
GPS
L1
C/A
8
D S.D. of DD carrier phase residuals (mm) 2.09
No. of SV
9
S.D. of DD carrier phase residuals (mm) 1.96
L2 L5 L1 GLONASS L2 BDS
B1 B2
L2C(M+L) L2P(Y) I+Q L1 OF L1 SF L2 OF L2 SF I I
6 8 2 7 7 7 7 6 6
2.76 2.44 2.37 2.16 1.80 2.69 2.12 3.58 2.70
6 9 2 7 7 7 7 6 6
2.83 2.39 2.61 2.35 1.83 2.95 2.20 3.40 2.59
Case 2 test results A summary of the positioning test results of Cases 2 to 6 is shown in Table 4. The test results of Case 2 show that using SIGMA-ε model can improve the positioning accuracy by 5% to 10%, but the positioning results are receiver dependent when using the SIMGA-ε. This is because the weight assignment of SIGMA-ε model is based on CNR, and the CNRs of Javad datasets are higher than those of Septentrio datasets. Though two receivers are connected to the same antenna via a signal splitter, their CNRs are not the same because of differences in band limiting and quantization schemes within the receivers [22]. Figure 6 shows CNRS of two selected satellites, PRN11 with an elevation of about 26 degrees and PRN27 with an elevation of 67 degrees, from Javad and Septentrio receivers for Dataset A when the same antenna is used via a splitter. As a result, SIGMA-ε model is more receiver dependent compared with simple elevation model in the datasets.
Table 4 Statistical results of RMS error in Case 2
Case
Configuration
Dataset
Receiver model
2 2 2 2
Elevation model Elevation model SIGMA-ε model SIGMA-ε model
A A A A
Javad Septentrio Javad Septentrio
RMS Error (mm) Horizontal 5.0 5.0 4.7 4.6
Vertical 6.3 6.3 6.5 5.8
3D 8.0 8.0 8.0 7.4
Fig. 6 CNRs of GPS Satellites PRN11 and PRN27 with Javad and Septentrio receivers in Dataset A. CNRs of Javad dataset are higher than Septentrio dataset collected with the same antenna.
Case 3 test results In Table 5, the test results of Dataset A in Case 3 show that 10 Hz sampling rate can improve positioning accuracy by about 16% because more samples are used for averaging in higher sampling rate. The positioning quality of Dataset B is significantly degraded compared with Dataset A; it is due to severe multipath effects and poor satellite geometry because several satellites were blocked by the building as shown Figure 4(d). With the determined trajectory of rotation, the ‘true’ carrier phase measurement errors of some selected satellites in Datasets A and B are estimated and plotted in Figure 7. Figure 7 shows that Dataset A has large multipath errors around Epoch 500 when the antenna moves close to the wall, and Dataset B has larger multipath errors (up to 4 cm) than Dataset A. Though long tests have been carried out, we choose one complete cycle of rig rotation because the multipath errors in next cycle are very close to the previous cycle.
Table 5 Statistical results of RMS error in Case 3
Case 3 3 3 3
Configuration 1 Hz sampling rate 10 Hz sampling rate 1 Hz sampling rate 10 Hz sampling rate
Dataset A A B B
Receiver model Javad Javad Javad Javad
RMS Error (mm) Horizontal 4.5 3.7 11.9 9.2
Vertical 7.2 6.1 34.0 17.2
3D 8.5 7.1 36.0 19.5
Fig. 7 ‘True’ GPS L1 double-differenced carrier phase measurement errors in Datasets A (left) and B (right).
Case 4-6 test results In Table 6, the results of Case 4 show that the dynamics of the antenna has no significant effect on positioning quality. The results of Case 5 in Table 6 show that the kinematic positioning with millimeter-level accuracy significantly outperforms decimeter-accurate PPP in the datasets. This can be seen from Figure 8, where most of the PPP solutions fall outside the true trajectory (red line) whilst kinematic positioning solutions are close to the true trajectory. The main sources of decimeter-level error of PPP in Figure 8 are due to the tropospheric delay, satellite orbit and
clock errors. Unlike PPP, these errors are cancelled out in relative positioning with double differenced measurements, hence the positioning accuracy of relative positioning is better than that of PPP. The results of Case 6 show that using multi-GNSS (GPS+GLOANSS+Galileo+ BDS+QZSS) can significantly improve the kinematic positioning accuracy by about 47% compared with using GPS only in a clear open area.
Table 6 Statistical results of RMS error in Cases 4-6
Case 4 4 5 5 6 6
Configuration Speed of rotation 0.72 m/s Speed of rotation 0.21 m/s Precise Point Positioning Kinematic mode GPS only Full constellations
Receiver model
RMS Error (mm) Horizontal
Vertical
3D
C
Javad
2.1
6.4
6.7
D
Javad
2.2
6.8
7.2
A A C C
Septentrio Septentrio Javad Javad
107.8 4.6 2.2 1.5
60.6 5.4 6.8 3.5
123.3 7.0 7.2 3.8
Dataset
Fig. 8 Positioning results (blue) and true trajectory (red) using PPP (top) and kinematic positioning (bottom)
Conclusions This work has developed a rotating arm rig for kinematic testing of GNSS carrier-phase based high precision positioning. The method for determination of the trajectory of rotation with a submillimeter-level accuracy has been presented. Four datasets were collected using the rig for testing in six cases. The double differenced carrier phase residuals of GPS, GLONASS, and BDS signals in kinematic testing have been assessed in Case 1. Statistical results show the residuals slightly vary with signal types, and most of the residuals are from 2.5 mm to 4 mm in kinematic conditions. Also, the results show that the dynamics of the antenna, i.e. moving speed of antenna, has no significant effect on signal quality. The test on stochastic models in Case 2 shows that SIGMA-ε model can improve the positioning accuracy by 5% to 10% compared with simple elevation model, but SIGMA-ε model is receiver-dependent since it assigns measurement weights based on CNR. The test results of Case 3 show that a higher sampling rate (10 Hz) can improve the positioning quality comparing with 1 Hz sampling rate, especially in a difficult environment. The results also show that with resolved ambiguity, carrier phase multipath errors of relative positioning are from few millimeters up to 4 centimeters. The test results of Case 4 show that the dynamics of the antenna has no significant effect on positioning quality. The results of Case 5 show that the kinematic positioning with millimeter-level accuracy significantly outperforms decimeter-accurate Precise Point Positioning in the dataset because some sources of
errors such as tropospheric delay are cancelled out in double differencing of relative positioning. The results of Case 6 show that using multi-GNSS (GPS+GLOANSS+Galileo+BDS+QZSS) can significantly improve the kinematic positioning accuracy by about 47% in a clear open area. This article has demonstrated the use of the proposed rotating arm rig for validation of GNSS data processing algorithms and error mitigation methods in kinematic conditions. Therefore, the main contribution of this research is the design of a low-cost, low power consumption, high precision solution for rigorous kinematic GNSS research validation and performance assessments. Based on the designed rotating arm rig, in the future it is possible to use the rig for the testing and validation of other technologies and algorithms. For example, replacing the prism and antenna used in this study with photogrammetric targets or Ultra-Wide Band signal transmitter, or study relevant machine learning and filtering methods [23,24,25] for multi-sensor data fusion.
Acknowledgements The cost of building the rotating arm rig is funded by Lawrence Lau’s internal research grant (01.03.02.01.2033).
References [1] Lau, L., Tateshita, H., Sato, K., 2015, Impact of Multi-GNSS(GPS, GLONASS, and QZSS) on positioning accuracy and multipath errors in high-precision single-epoch solutions - A case study in Ningbo China. J. Navig. 68 (5) 999-1017. https://doi.org/10.1017/S0373463315000168. [2] Quan, Y., Lau, L., Roberts, G.W., Meng, X., 2015, Measurement signal quality assessment on all available and new signals of multi-GNSS (GPS, GLONASS, Galileo, BDS, and QZSS) with real data. J. Navig. 69 (2) 313-334. https://doi.org/10.1017/S0373463315000624. [3] Kechine, M.O., Tiberius, C.C.J.M., Van der Marel, H., 2004, An experimental performance analysis of real-time kinematic positioning with NASA’s Internet-based global differential GPS. GPS Solut. 8 (1) 9-22. https://doi.org/10.1007/s10291-003-0081-3.
[4] Sohn, S., Lee, B., Kim, J., Kee, C., 2008, Vision-based real-time target localization for single-antenna GPS-guided UAV. IEEE Trans. Aerosp. Electron. Syst. 44 (4) 1391-1401. https://doi.org/10.1109/TAES.2008.4667717. [5] Lau, L., Cross, P., Steen, M., 2012, Flight tests of error-bounded heading and pitch determination with two GPS receivers. IEEE Trans. Aerosp. Electron. Syst. 48 (1) 388-404. https://doi.org/10.1109/TAES.2012.6129643. [6] Wang, D., Lv, H., Wu, J., 2017, Augmented Cubature Kalman filter for nonlinear RTK/MIMU integrated navigation with non-additive noise. Measurements 97 111-125. https://doi.org/10.1016/j.measurement.2016.10.056. [7] Meng, X., Liu, C., 2013, Precise determination of mini railway track with ground-based laser scanning. Surv. Rev. 46 (336) 213-218. https://doi.org/10.1179/1752270613Y.0000000072. [8] Betaille, D., Cross, P., Euler, H.J., 2006, Assessment and improvement of the capabilities of a window correlator to model GPS multipath phase errors. IEEE Trans. Aerosp. Electron. Syst. 42(2) 707-718. https://doi.org/10.1109/TAES.2006.1642583. [9] Lau, L., Cross, P., 2007, Investigations into phase multipath techniques for high precision positioning in difficult environments. J. Navig. 60 (3) 457-482. https://doi.org/10.1017/S0373463307004341. [10] Sun, Q., Xia, J., Foster, J., Falkmer, T., 2017, Pursuing precise vehicle movement trajectory in urban residential area using Multi-GNSS RTK tracking. Transp. Res. Procedia. 25:23612376. https://doi.org/10.1016/j.trpro.2017.05.255. [11] Allan, A.L., 1996, Circle fitting in three dimensions. Surv. Rev. 33 (260) 416-422. https://doi.org/10.1179/sre.1996.33.260.416. [12] Al-Sharadqah, A., Chernov, N., 2009, Error analysis for circle fitting algorithms. Electron. J. Stat. 3(2009) 886-911. https://doi.org/10.1214/09-EJS419. [13] Kasa, I., 1976, A curve fitting procedure and its error analysis. IEEE Trans. Inst. Meas. 25(1) 8-14. https://doi.org/10.1109/TIM.1976.6312298. [14] Leick, A., 2004, GPS Satellite Surveying. 3rd Edition, John Wiley and Sons, Hoboken. ISBN: 978-0-471-05930-1
[15] Chernov, N., Lesort, C., 2005, Least squares fitting of circles. J. Math. Imag. Vision 23 239–251. https://doi.org/10.1007/s10851-005-0482-8. [16] Pratt, V., 1987, Direct least-squares fitting of algebraic surfaces. Comput. Graphics 21(4) 145-152. https://doi.org/10.1145/37402.37420. [17] Taubin, G., 1991, Estimation of planar curves, surfaces and nonplanar space curves defined by implicit equations, with applications to edge and range image segmentation, IEEE Trans. Pattern Anal. Mach. Intell. 13 (11) 1115–1138. https://doi.org/10.1109/34.103273. [18] Uren, J., Price, B., Surveying for Engineers, fifth ed., Red Globe Press, London, 2010. [19] Jolliffe, I., 2002, Principal component analysis, 2nd ed. Springer. ISBN 978-0-387-22440-4 [20] Hartinger, H., Brunner, F.K., 1999, Variances of GPS phase observations: the SIGMA-ε model. GPS Solut. 2(4) 35-43. https://doi.org/10.1007/PL00012765. [21] Teunissen, P.J.G., 1995, The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J. Geod. 70(1-2) 65-82. https://doi.org/10.1007/s001900050127. [22] Joseph, A., 2010, Measuring GNSS signal strength. Inside GNSS 5(8):22-25. [23] Akbarizadeh, G., 2012, A new statistical-based kurtosis wavelet energy feature for texture recognition of SAR images. IEEE Trans. Geosc. Remote Sens. 11(50) 4358-4368. https://doi.org/10.1109/TGRS.2012.2194787. [24] Karimi, D., Akbarizadeh, G., Rangzan, K., Kabolizadeh, M., 2017, Effective supervised multiple-feature learning for fused radar and optical data classification. IET Radar Sonar Navig. 11(5) 768-777. https://doi.org/10.1049/iet-rsn.2016.0346. [25] Norouzi, M., Akbarizadeh, G., Eftekhar, F., 2018, A hybrid feature extraction method for SAR image registration. Signal Image Video Process. 12: 1559. https://doi.org/10.1007/s11760-018-1312-y.
Highlights
A rotating arm rig with one end hosting a prism or a GNSS antenna is proposed True circular trajectory of arm rotation is mathematically determined by total station measurements Determined trajectory served as a reference in kinematic GNSS positioning testing Assessing signal quality and positioning quality in kinematic condition Study effects of stochastic model, sampling rate, antenna moving speed, and GNSS constellation